3 Answers
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See this comprehensive survey at Aish, about plastic surgery in general. It mentions this kohen reason, in the name of Rabbi Menashe Klein, in his Mishneh Halachos, and (IIUC) Rav Shlomo Zalman Aurbach. But others permit for very different reasons:

Thus:

In 1961, Rabbi Immanuel Jakobovits, considered by many to be the father of the discipline of Jewish medical ethics, 3 addressed the American Society of Facial Plastic Surgery at a symposium entitled "Religious Views on Cosmetic Surgery."... He concluded6 definitively that plastic surgery for aesthetic enhancement is a form of arrogance and vanity (particularly for men) and is forbidden unless the patient meets certain criteria. He later wrote as part of an overview of the Jewish approach to medicine:

In the sparse rabbinic writings on the subject, these reservations could be discounted, provided the danger is minimal; and especially 1) if the operation is medically indicated, e.g. following an accident, or for grave psychological reasons; 2) if the correction of the deformity is designed to facilitate or maintain a happy marriage; or 3) if it will enable a person to play a constructive role in Society and to earn a decent livelihood. 7

Then, a bit later:

In 1964, ... Rabbi Mordechai Yaakov Breish, author of the Chelkas Yaakov and a prominent posek [authority in Jewish law] in Switzerland, discussed the issues of risk and chavala (self-injury) when asked whether a woman may undergo cosmetic surgery to straighten and decrease the size of her nose in order to improve her chance of finding a suitable husband. 10

He ruled in favor of allowing her to do this. In part,

Citing the psychological pain associated with the inability to find a spouse, Rabbi Breish ruled that the woman may have the cosmetic surgery.

A bit later,

The same year, Rabbi Moshe Feinstein (1895-1986) was asked the same question... His final ruling permits surgery when it is in the best interests of the patient, even if they are not sick and it does not treat an illness. As a result, he permitted the woman to have cosmetic surgery since it was to her advantage and not being done to harm her.

...

Also in 1964, Rabbi Menasheh Klein, author of Mishneh Halachos, dealt with the question of the permissibility of cosmetic surgery to correct various facial imperfections that mar a woman's appearance, such as a very long nose which makes it difficult for her to marry and which she feels makes her very unattractive.

The Mishna22 discusses the case of a man who betroths a woman on the condition that she has no defect (mum) where a "mum" is defined as any defect that would bar a Cohen (Jewish priest) from serving in the Temple. Tosofos23 states that if the woman had her blemish corrected by a physician before her engagement, the marriage is valid. Since many of the blemishes that would apply to a Cohen include cosmetic imperfections24 of the face for which people today would desire elective plastic surgery and Tosofos permits these blemishes to be corrected by a physician, Rabbi Klein states that it appears that a man or woman may go to a doctor to correct a cosmetic defect merely for enhancement of their appearance. Rabbi Klein rejects the argument that plastic surgery entails any danger whatsoever based the information which he received from physicians.

There is also a position that it is OK except for the sakanah involved in it:

In 1967, Rabbi Yitzchak Yaakov Weiss (1902-1989), head of the Eida Chareidis rabbinical court in Jerusalem and author of Minchas Yitzchak, dealt briefly with the issues of chavala and risk with respect to plastic surgery. 26 He takes the same approach to self-injury as Rabbi Feinstein, arguing that the prohibition of chavala only applies when the wound is inflicted with the intention of causing harm or degradation. He feels that cosmetic surgery would be permitted if not for the risk of surgery, which he believes to be a serious concern.

A halachic position against plastic surgery, including hashkafically:

This is one of the major concerns voiced by Rabbi Eliezer Yehuda Waldenberg, author of Tzitz Eliezer, a multivolume set of responsa, much of which deals with medical issues. First, Rabbi Waldenberg29 objects to performing surgery on someone who is neither sick nor in pain. 30 He argues that such activities are outside the boundaries of the physician's mandate to heal (since he questions whether cosmetic surgery is truly included in the category of healing). He further asserts that the patient has no right to ask the physician to wound him or her for the purposes of merely enhancing beauty. Rabbi Waldenberg then makes the theological argument that as the ultimate artisan, God creates each person in His image, exactly as he or she should be, with nothing extra nor anything lacking. He therefore posits that cosmetic surgery that is not for pain or true illness is an affront to God and is forbidden.

And finally, Rav Shlomo Zalman Aurbach:

if the plastic surgery is done to prevent suffering and shame caused by a defect in his looks (for instance a nose which is very abnormal) this would be permitted based on the Tosafot and the Gemara, since the purpose is to remove a blemish. However if the only reason is for beauty, this is not permitted.

For precise sources, follow the link to the article and look at the footnotes.

Just to add to the many valid opinions already brought down, Hacham Ovadia Yosef (Yabia Omer C"M 8:12) permits all forms of plastic surgery for both married and unmarried women and men, as long as they do it by a professional to minimize the risk.

Funny that today we all think about the need for "nose jobs" and cosmetic surgery for noses that are too long; in theory, a nose too short is a disqualification for a Cohen too! For a less vain example, let's talk about the case of the crooked nose (also on the list).

Rabbi Moshe Feinstein had allowed rhinoplasty for young woman who felt that her nose was seriously hurting her efforts to find a suitor. It has to do with when one is allowed to have injury caused to one's self (not to mention risk) for valid reasons.

(From here on out is my own reasoning, not Rabbi Feinstein's.)

The problem here becomes -- as America today is beginning to realize -- that we can get these very distorted senses of what's "normal" and not (e.g. based on the photoshopped faces we see on magazine covers), and before you know it, everyone "needs" a nose job. (What about this case of the newlywed groom who storms into the rabbi's office, furious that he doesn't like the look of his bride's private parts. Is the correct response she may have cosmetic surgery for the sake of their marriage, or is it slap this kid upside the head, yes this is what a real human being looks like so get over it). This is more of a philosophical question.

Note that Judaism believes in the use of science and technology to cure diseases. So we'd be inclined to say that if someone is actually disfigured, by all means use plastic surgery to "fix" them back to "normal." If someone was born with a perfectly fine, healthy nose that isn't quite the latest style these days, well then better to have some faith and be satisfied with one's lot.

One line that could be proposed is, as mentioned above, based on the list of disqualifying blemishes for a Kohen. If a woman was hiding one of these blemishes and she met a man who married her and thus agreed to paying her Ketubah, upon discovery of the blemish he could void his monetary offer. So it could be suggested that anything on the list (or equally severe) is something that medicine can fix (if we feel it needs fixing! If she was born without eyebrows but both he and she like the look, that's fine too). To try and fix anything else would be hubris.

As such I'd wonder if, in instances where plastic surgery is indicated, the goal should be an "averagely-proportioned" nose, not a "Hollywood photoshop nose." It's amazing how all the people on these plastic-surgery makeover shows wind up looking alike after the procedures.

+1. About the "long nose disqualifies a kohen" - it's in Rambam, Bias Mikdash 8:7 (from Bechoros 44a): the expected proportion is that the nose should be the same length as the little finger.
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AlexOct 11 '11 at 2:03

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Shalom based on your eror I would say you should edit your answer
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ChalutzhanalOct 11 '11 at 2:15